The following interesting
explanation of a Diffie-Hellman key exchange
, including a 2-line (!
) Perl script
to implement it, is from:
It is by Adam Back. (Munge
d email: <not eve but her partner ____> at cypherspace dot org).
that I had to fudge
a lot of things in it (some <pre> tags
wanted to treat < as opening an HTML tag
, and some didn't.) So if there's anything weird
, lemme' know. (Unless I've just
noded this...I have a sneaking suspicion
that the Scratch Pad
isn't quite WYSIWYG
...in which case, I am already busily working to fix it!).
Diffie-Hellman in 2 lines of perl
#!/usr/local/bin/perl -- -export-a-crypto-system-sig Diffie-Hellman-2-lines
($g,$e,$m)=@ARGV,$m||die"$0 gen exp mod\n";print`echo "16dio1[d2%Sa2/d0<X+d
Diffie-Hellman key exchange allows two parties who have not met to
exchange keys securely on an unsecure communication path. Typically
D-H is used to exchange a randomly generated conventional encryption
key, the rest of the exchange is then encrypted with the conventional
cipher. It has been used with DES, 3DES, IDEA, RC4 though basically
the approach of using D-H key exchange can be used for any
conventional stream or block cipher.
PGP itself operates in a similar fashion, except that PGP uses RSA for
key exchange, and IDEA as the conventional cipher.
The maths for Diffie-Hellman is quite simple.
Here is an example:
We are trying to exchange a random key for communication. Say that we
will be using the RC4 stream cipher as our conventional cipher.
Here's the stages in the process.
You advertise your public generator g, and public prime modulus m as
your "public key" in the same way that you would advertise an RSA
public key, you give this to anyone you wish to exchange messages
with. This is no requirement for these numbers to be kept secret, all
knowledge of these numbers gives is the ability to send you messages.
You do not tell anyone what your secret exponent x is.
You initiate a key exchange by calculating a:
a = g mod m
and sending me the number a.
You can use the dh perl program to do this calculation. For the
sake of argument we will choose some numbers. Say that your public
generator g = 3, your public prime modulus m = 10001 and your secret
exponent x = 9a2e (all numbers in hex). Then you would run the perl
dh program like this to calculate a:
% dh g x m
You send me the following in email:
g = 3, m = 10001, a = c366
Now I choose a session key to use for RC4, using D-H key exchange
I can send this to you without an eavesdropper being able to get the
I do not choose a session key directly but rather choose a random
number y from which I will calculate the session key.
Say that the random y = 4c20.
Now I calculate the session key s:
s = a mod m
Or using the perl dh program:
% dh c366 4c20 10001
So the session key s = ded4
In order to send you the session s key I need only calculate:
b = g mod m
Again using the perl dh program:
% dh 3 4c20 10001
So b = 6246
I send the number b to you in email, now you can recover the session
s = b mod m
So to obtain the session key, using the perl dh program you do:
% dh 6246 9a2e 10001
And sure enough out pops the session key as I calculated, s = ded4.
Now we both have the session key and commence exchanging messages
using our agreed key as the key to a symmetric cipher such as rc4.
So now I can send you (or you can send me) rc4 messages encrypted with
our negotiated session key s = ded4:
% cat msg | rc4 ded4 | uuencode r r | mail
A perl (and a C version) of rc4 suitable for the above can be found
This works because of the mathematical property that:
g mod m = g mod m
a = g mod m, b = g mod m
=> s = a mod m = b mod m
x and y never change hands, and yet s has changed hands. Further you
can't discover my y and I can't discover your x, and an eavesdropper
has neither x nor y and so cannot discover s.
You can try sending me some RC4 encrypted email, using a D-H
negotiated session key.
For security (the above example is for clarity only 32 bit keys are
utterly useless for security purposes) we will use 1024 bits. Here is
my Diffie-Helman public key, you can have a go at negotiating a D-H key
exchange and sending me some RC4 encrypted email.
I have chosen an x (which I won't be telling you this time for obvious
reasons), and calculated the corresponding a as described above. Here
are my D-H public key numbers:
g = 3
To send me some mail you will need to generate your choice of random y
as described above, and then calculate the session key s:
% dh [a] [y] [m] > s
(where [a] and [m] are the large numbers above cut and pasted in, and
[y] is your large random number.)
and then calculate key exchange number b:
% dh [g] [y] [m] > b
% mail email@example.com < b
Then you compose your mail message as file "msg" and rc4 encrypt that
using s as calculated above:
% cat msg | rc4 `cat s` | uuencode r r | mail firstname.lastname@example.org
Comments, html bugs to me [contact info. as at top of node.]